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\title{Figures in Appendix B
}

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\begin{document}
\maketitle
\date{}



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%{\it\Large\bf(For Online Publication)}
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\section{Simulations}\label{apx:simu}
\begin{figure}
	% \centering
	\caption{}
	\resizebox {8cm} {5cm} {  
		\begin{tikzpicture}
			\node[below]at(0,0){};
			\draw[->] (0,0) -- (0,6) node[left]at(0,5.8){S};
			\draw[->] (0,0) -- (10,0)  node[right] at (10,0){$t$};
			\draw[-] (-0.1,0.5) -- (0.1,0.5) node[left]at(-0.1,0.5){$\tilde{S}$};
			\draw[-] (0.5,-0.1) -- (0.5,0.1) node[above]at(0.75,0){$t^a$};
			
			\draw[green] (0.5,0.5)--(7.2,4.85);
			%\draw[green,dashed] (0.5,0.5)--(6.5,7);
			%\draw[green,dashed] (0.5,0.5)--(6.5,3);
			
			\draw[rounded corners,red] (0.5,3.3) to (1.5,3.77) to (2.5,4.3);
			\draw[blue] (2.5,4.3)--(5.5,4.8)--(7.5,5);
			
			\draw[dashed] (2.5,4.35) -- (2.5,0) node[above]at(2.3,0){$\hat{t}$};
			\draw[dashed] (5.5,4.8) -- (5.5,0) node[above]at(5.3,0){$t^m$};
			
			\draw [decorate,decoration={brace,mirror,amplitude=10pt}]
			(0.5,-0.2) -- (6.5,-0.2)node [midway,align=center,yshift=-25pt] {$e=h$};  
		\end{tikzpicture}
	}
	\floatfoot{{\it Notes}: This figure represents the scenario that the ruler's health deteriorates quickly such that the honeymoon phase is connected with the power transition phase directly. This case indicate that the peaceful power transition is highly possible or the successor challenges the ruler with a high chance to win. The red curve represents the change of the monitoring thresholds $\bar{s}^m_t$ with time. The blue curve represents the change of the challenge thresholds $\bar{s}^c_t$ with time. When $t\leq\hat{t}$, $\bar{s}^m_t=\bar{s}^c_t$; when $\hat{t}<\bar{t}^m$, $\bar{s}^c_t<\bar{s}^m_t$; when $\bar{t}^m\leq t$, $\bar{s}^m_t$ does not exists. The green line represents the change of the expected power accrued by the successor with time. Since the successor always chooses high effort, the average power increase rate is $p_h(H-L)+L$. The parameters are chosen as follows: $b=10$, $\delta=0.7$, $R=10$, $r=0.1$, $L=0.001$, $H=0.01$, $p_h=0.5$, $w=0.05$, $\eta=0.7$ $p_t=p_{t-1}+0.01$, $p_0=0$, $\tilde{S}=0.01$. Thresholds are calculated as: $\hat{t}=6$, $t^m=60$.
		
	} 
	\label{fig: long honeymoon}  
\end{figure}


\begin{figure}
	% \centering
	\caption{}
	\resizebox {8cm} {5cm} {  
		\begin{tikzpicture}
			\node[below]at(0,0){};
			\draw[->] (0,0) -- (0,6) node[left]at(0,5.8){S};
			\draw[->] (0,0) -- (10,0)  node[right] at (10,0){$t$};
			\draw[-] (-0.1,0.5) -- (0.1,0.5) node[left]at(-0.1,0.5){$\tilde{S}$};
			\draw[-] (0.5,-0.1) -- (0.5,0.1) node[above]at(0.75,0){$t^a$};
			
			\draw[green] (0.5,0.5)--(4.5,4)--(6,4.4)--(6.6,4.8);
			%\draw[green,dashed] (0.5,0.5)--(6.5,7);
			%\draw[green,dashed] (0.5,0.5)--(6.5,3);
			
			\draw[rounded corners,red] (0.5,2.7) to (3,3.7) to (6,4.7);
			\draw[blue] (6,4.7)--(7.5,5.15);
			
			\draw[dashed] (6,4.7) -- (6,0) node[above]at(5.8,0){$\hat{t}$};
			\draw[dashed] (7.1,5) -- (7.1,0) node[above]at(6.87,0){$t^m$};
			
			\draw [decorate,decoration={brace,mirror,amplitude=10pt}]
			(0.5,-0.2) -- (4.5,-0.2)node [midway,align=center,yshift=-25pt] {$e=h$}; 
			\draw [decorate,decoration={brace,mirror,amplitude=10pt}]
			(4.5,-0.2) -- (6,-0.2)node [midway,align=center,yshift=-25pt] {$e=l$}; 
			\draw [decorate,decoration={brace,mirror,amplitude=10pt}]
			(6,-0.2) -- (7,-0.2)node [midway,align=center,yshift=-25pt] {$e=h$};
		\end{tikzpicture}
	}
	\floatfoot{{\it Notes}: This figure represents the scenario that the ruler's health deteriorates moderately such that after the honeymoon phase the successor has to keep a low profile to avoid the conflict and wait for the deterioration of the ruler's health. This situation indicates that the successor has a high chance to succeeds the throne.	The red curve represents the change of the monitoring thresholds $\bar{s}^m_t$ with time. The blue curve represents the change of the challenge thresholds $\bar{s}^c_t$ with time. When $t\leq\hat{t}$, $\bar{s}^m_t=\bar{s}^c_t$; when $\hat{t}<\bar{t}^m$, $\bar{s}^c_t<\bar{s}^m_t$; when $\bar{t}^m\leq t$, $\bar{s}^m_t$ does not exists. The green line represents the change of the expected power accrued by the successor with time. When the successor chooses high effort, the average power increase rate is $p_h(H-L)+L$. When the successor chooses low effort, the power increase rate is $L$. The parameters are chosen as follows: $b=10$, $\delta=0.7$, $R=10$, $r=0.1$, $L=0.001$, $H=0.01$, $p_h=0.5$, $w=0.05$, $\eta=0.7$ $p_t=p_{t-1}+0.005$, $p_0=0$, $\tilde{S}=0.29$. Thresholds are calculated as: $\hat{t}=81$, $t^m=119$.
	} 
	\label{fig: no conflict} 
\end{figure}

\begin{figure}
	%   \centering
	\caption{}
	\resizebox {8cm} {5cm} {  
		\begin{tikzpicture}
			
			\node[below]at(0,0){};
			\draw[->] (0,0) -- (0,6) node[left]at(0,5.8){S};
			\draw[->] (0,0) -- (10,0)  node[right] at (10,0){$t$};
			\draw[-] (-0.1,0.5) -- (0.1,0.5) node[left]at(-0.1,0.5){$\tilde{S}$};
			\draw[-] (0.5,-0.1) -- (0.5,0.1) node[above]at(0.75,0){$t^a$};
			
			\draw[green] (0.5,0.5)--(3,3.5)--(4.1,3.85)--(4.35,4.1);
			%\draw[green,dashed] (0.5,0.5)--(6.5,7);
			%\draw[green,dashed] (0.5,0.5)--(6.5,3);
			
			\draw[rounded corners,red] (0.5,2.7) to (3,3.7) to (6.5,4.9);
			\draw[blue] (6.5,4.9)--(8.5,5.3);
			
			\draw[dashed] (6.5,4.9) -- (6.5,0) node[above]at(6.2,0){$\hat{t}$};
			\draw[dashed] (6.7,4.9) -- (6.7,0) node[above]at(7,0){$t^m$};
			
			\draw [decorate,decoration={brace,mirror,amplitude=10pt}]
			(0.5,-0.2) -- (4.5,-0.2)node [midway,align=center,yshift=-25pt] {$e=h$}; 
			\draw [decorate,decoration={brace,mirror,amplitude=10pt}]
			(4.5,-0.2) -- (6,-0.2)node [midway,align=center,yshift=-25pt] {$e=l$}; 
			\draw [decorate,decoration={brace,mirror,amplitude=10pt}]
			(6,-0.2) -- (7,-0.2)node [midway,align=center,yshift=-25pt] {$e=h$};
			
		\end{tikzpicture}
	}
	\floatfoot{{\it Notes}: This figure depicts the scenario under which a ruler remains healthy for a relatively long time. 
		This situation indicates that the successor has a high chance to conflict with the ruler and low chance to win. The red curve represents the change of the monitoring thresholds $\bar{s}^m_t$ with time. The blue curve represents the change of the challenge thresholds $\bar{s}^c_t$ with time. When $t\leq\hat{t}$, $\bar{s}^m_t=\bar{s}^c_t$; when $\hat{t}<\bar{t}^m$, $\bar{s}^c_t<\bar{s}^m_t$; when $\bar{t}^m\leq t$, $\bar{s}^m_t$ does not exists.
		The parameters are chosen as follows: $b=10$, $\delta=0.7$, $R=10$, $r=0.1$, $L=0.001$, $H=0.01$, $p_h=0.5$, $w=0.05$, $\eta=0.7$ $p_t=p_{t-1}+0.001$, $p_0=0$, $\tilde{S}=0.01$. Thresholds are calculated as: $\hat{t}=404$, $t^m=592$.
	}  
	\label{fig: conflict}  
\end{figure} 

  


\end{document}
